Satisfiability solving is a common technique for formal verification forming the basis of many proof and model checking systems. Failure to show a proof obligation will produce a counterexample or failure trace with typically many thousands or even millions of boolean variables. Interpreting such a counterexample poses a challenge. Even if the individual variables are all understood, it is difficult to form a "big picture" of the situation causing the failure. We consider the case where verification conditions are expressed using concepts from a formal application domain model in a language based on predicate logic or a similar language. We introduce a method to explain verification failures in application domain terms. A measure of the relative relevance of predicates is used to extract the parts of a formula most likely to contribute meaningfully to the explanation. Dependencies between predicates are used to form a branching sequence of successive explanations. These explanations can help a practitioner find faults in the system being verified. The method is demonstrated on examples and compared to other methods.

We consider additive functionals $X_n(ϕ)$ with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if $s_0=s_1=0$ and for some models with Beta-distributed splitter. For $s_0+s_1>0$, the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method.

Crossing symmetry suggests that deep inelastic scattering and semi inclusive electron-positron annihilation are governed by analytic continuations of a single forward amplitude. Drell, Levy, and Yan proposed that the hadronic tensor admits analytic continuation and demonstrated, in reasonable models, that connected contributions to the cross-section continue. They also identified obstructions to continuation of the current correlator. In this work we supplement their observation with a new dispersive proposal for analytic continuation of the correlator and, assuming polynomial boundedness, derive subtracted dispersion relations relating spacelike and timelike cross sections. We introduce a new factorized function that quantifies the obstruction to crossing and compute its hard kernel at lowest order. The resulting identity connects distribution functions in deep inelastic scattering to fragmentation functions in annihilation.

In this work, we study the stabilization of the wave equation using an internal delayed potential. Interestingly, the stabilization mechanism is entirely induced by the delay, since exponential stabilization cannot be achieved in its absence. We first prove the well-posedness of the associated initial--boundary value problem. Then, thanks to the parametric analysis of the corresponding quasipolynomial, we design a delayed po tential feedback law which, together with appropriate initial conditions, ensures the exponential decay rate for the resulting closed-loop system. The control of the transverse vibration of a string illustrates the effectiveness of the result.

Cross-cutting commenting on social media is often imagined as a path to deliberation, yet exposure to opposing views frequently fuels hostility. To explain this dynamic, we introduce the concept of partisan warriors--commenters who cross ideological lines primarily to launch uncivil attacks against out-partisans. We analyze a large corpus of YouTube comments (N= 1,854,320) surrounding the 2024 U.S. second presidential debate. After filtering for toxicity and active participation, we use large language models to identify attack targets and operationalize partisan warrior behavior. Our analysis highlights four dynamics. First, cross-cutting commenters do not exhibit greater civility than those who remain within their ideological camps (RQ1). Second, audience reactions diverge by ideology: conservative audiences tended to reward hostile attacks on out-group leaders, whereas liberal audiences offered no comparable incentives and at times penalized such attacks (RQ2). Third, partisan warriors are notably more prevalent in conservative-leaning channels than in liberal ones; commenters restricted to conservative spaces were substantially more likely to engage in partisan warrior behavior compared to their liberal-only counterparts (RQ3). Finally, regarding environmental triggers, robustness checks suggest that this participation is an ecological phenomenon driven largely by channel-level heterogeneity rather than transient responses to individual video titles (RQ4). By shifting attention from the prevalence of incivility to its targets, rewards, and structural drivers, this study advances understanding of how partisan hostility is enacted and sustained in online spaces.

In this paper we discuss the multiplicity distribution in the deep inelastic processes in the frame work of high energy QCD. We obtained three results. First, we get the new derivation of the equations for the cross sections of productions of $n$-cut Pomerons in the final states ($σ_n$). These equations coincide with the equations that have been derived using the Abramovsky, Gribov and Kancheli (AGK) cutting rules but based on the dipole approach to QCD. Second, we developed the homotopy approach for finding the solutions to these equations. It consists with the analytic solution for the first iteration and the converge procedure of calculating the next iterations using computing. Third, we found the analytical solution for $σ_n$ at large $n\,\gtrsim\,N(z) = 2 N_0 \,z\,\exp( z^2/(2 κ))$ with $z = \ln( r^2\,Q^2_s )$. Using this solution we calculate the entropy of the produced gluons at large $z$: $S_E = \ln \left( N(z)\right)$, where the saturation momentum $Q_s$ and all constants are discussed in the text.

This paper is devoted to the definition and analysis of the spectral shift function (SSF) associated with non-self-adjoint perturbations of self-adjoint operators. Motivated by applications in scattering theory, we consider both trace-class and relatively trace-class perturbations. We extend the Lifshits-Kre__n trace formula to non-self-adjoint operators under suitable assumptions on the spectrum and the behavior of the resolvent. The role of spectral singularities is carefully analyzed, and we provide a generalization of the SSF using functional calculus. Finally, we apply our results to Schr{ö}dinger operators with complex-valued short-range potentials in dimension three. Toy models illustrate properties that one might hope to extend to general cases. In particular, they suggest that the SSF carries information on the presence of complex eigenvalues.

Energy efficiency has become a growing concern in software development, leading to the need for tools designed to measure energy consumption. While several energy measurement tools are available as open-source projects, their characteristics and adoption remain underexplored. This work presents an empirical study based on a Mining Software Repositories (MSR) approach to identify, classify, and analyze software energy monitoring tools publicly available on GitHub. We qualitatively analyzed an initial dataset of 585 repositories to identify key design aspects, including measurement granularity and underlying design principles. After this analysis, we retained 24 repositories as relevant energy measuring software tools. The qualitative analysis we conduct reveals a clear evolution from early CPU-centric and machine-level monitoring utilities toward more diverse tools that support multi-level granularity (process, container, and AI workload levels) and integrate emission estimation capabilities. This study provides the first structured overview of open-source energy and emission measurement tools from an MSR perspective, which may be beneficial for software architects when designing energy-aware software.

Rigorous, non-asymptotic bounds for the Puiseux expansion of the eigenvalue at infinity are given. Error analysis is provided. Further, the expected value of the eigenvector condition number of a randomly perturbed matrix is estimated. The latter result is applied to the Cayley transform of the linear pencil. Numerical simulations illustrating the theoretical findings are provided.

Recent studies have identified materials and devices whose behavior lies beyond the scope of conventional electronic-structure theory. Such theories are formulated entirely in terms of Hamiltonian evolution and therefore describe only unitary dynamics and thus only a restricted class of quantum systems. In contrast, electron systems that incorporate quantum measurement as an intrinsic dynamical element undergo Hamiltonian evolution interleaved with projection-induced state updates. This unitary-projective dynamics breaks constraints imposed by purely unitary evolution and permits stochastic population transfer between symmetry-related transport channels, thereby enabling fundamentally new material functionalities. This insight motivates the deliberate design of materials and devices that harness unitary-projective dynamics. This article explores the foundations of unitary-projective electron dynamics and charts the resulting landscape of quantum materials and their functionalities. Model calculations demonstrate passive mesoscopic structures with intrinsic nonreciprocal single-electron transmission, materials exhibiting a novel category of magnetism, and possible platforms for energy harvesting and conversion with efficiencies that exceed the standard Carnot limit.

The evolutionary role of the so-called ``changing-look'' (CL) active galactic nucleus (AGN), which is characterized by spectral-type transitions within $\sim10$ yr, has been suggested in the past few years. By focusing on CL-AGNs having spectra similar to those of broad-line Seyfert 1 galaxies, some authors have proposed that CL-AGNs tend to be at a special evolutionary stage associated with intermediate-to-old stellar populations. Here we attempt to verify this evolutionary role by extending the sample to CL narrow-line Seyfert 1 (NLS1) galaxies, which are believed to be ``young'' AGNs with a less massive supermassive black hole and high accretion rate. Combining the recent large NLS1 catalog provided by Paliya et al. (2024) and the SDSS-V DR19 spectral survey returns only three CL-NLS1s out of a parent sample of 884 objects, reinforcing the rarity of CL-NLS1s. Subsequent spectral analysis shows that the evolutionary role mentioned above still holds, although CL-NLS1s tend to occupy the young end of the intermediate-old population. Finally, we propose that off-center SDSS spectra caused by the ``fiber drop'' effect have great potential for determining the properties of the narrow-line region of NLS1s.

This paper is accountable only to explicitly stated physical assumptions and strict logical inference. Its goal is to run a rigorous stress test of second-law claims within the Clausius framework. We work directly with \textbf{Clausius's entropy definition} for an isolated composite with energy-form conversion. Heat is withdrawn from a cold releasing subsystem with relatively small heat capacity, converted to electrical energy, and then delivered as heat to a hotter subsystem. In the ideal limit, the electrical leg contributes negligibly to Clausius entropy accounting, so the modeled reservoir Clausius sum is \[ ΔS_{\mathrm{Cl}} = Q\!\left(\frac{1}{T_B}-\frac{1}{T_A}\right) < 0. \] The paper provides a derivation, numerical illustrations, and a scope analysis; any claimed contradiction should be interpreted as a compatibility issue between different axiom sets, not as an algebraic error in the Clausius bookkeeping above.

We present a systematic simulation study of identified hadron production in minimum bias Au+Au collisions at sqrt(sNN) = 6, 7, and 8 GeV. The event samples were generated with three modern frameworks based on different microscopic pictures: EPOS-LHC, EPOS-4, and the purely hadronic transport model SMASH. We compare observables that probe baryon stopping, transverse dynamics, hadron formation, and strangeness production: rapidity densities dN/dy, transverse momentum spectra dN/dpT, two dimensional pT-y distributions, v2/nq versus pT/nq, and the yield ratios pi-/pi+, K-/K+, pbar/p, K+/pi+, K-/pi-, p/pi+, and Lambda/pi+. For charged and neutral pions, the three models give broadly similar yields and spectral shapes in both dN/dy and dN/dpT. At these energies, resonance decays and isospin constraints reduce the sensitivity to early stage dynamics. In contrast, strange mesons and baryons remain strongly model dependent. EPOS-4 gives the largest midrapidity K+ and Lambda yields and the hardest kaon and strange baryon pT spectra. EPOS-LHC is generally intermediate, while SMASH gives lower strange hadron production. The two dimensional pT-y maps show that the EPOS models populate higher pT over a broader rapidity range. For NCQ scaled elliptic flow, the best approximate scaling is seen in EPOS-LHC, while SMASH and EPOS-4 show only partial scaling. This suggests that EPOS-LHC carries a more coherent partonic anisotropy to the hadronic stage in this energy range. Overall, the model separation grows from 6 to 8 GeV. The clearest discriminators in the NICA domain are intermediate pT baryon to meson ratios, Lambda/pi+, the rapidity dependence of pT, the pT dependence of K-/K+, and NCQ scaled v2.

Shifting travel from private cars to public transport is critical for meeting climate and related mobility goals, yet passengers will only choose transit if it offers a consistently positive experience. Previous studies of passenger satisfaction have largely relied on retrospective surveys, which overlook the dynamic and spatially differentiated nature of travel experience. This paper introduces a novel combination of real-time experience sampling and spatial hot spot analysis to capture and map where public transport users report consistently positive or negative experiences. Data were collected from 239 participants in Hamburg between March and September 2025. Using a smartphone application, travelers reported their momentary journey experience every five minutes during everyday trips, yielding over 21,000 in-situ evaluations. These geo-referenced data were analyzed with the Getis-Ord $Gi^{*}$ statistic to detect significant clusters of positive and negative travel experience. The analysis identified distinct hot and cold spots of travel experience across the network. Cold spots were shaped by heterogeneous problems, ranging from predominantly delay-dominated to overcrowding or socially stressful locations. In contrast, hot spots emerged through different pathways, including comfort-oriented, time-efficient or context-driven environments. The findings highlight three contributions. First, cold spots are not uniform but reflect specific local constellations of problems, requiring targeted interventions. Second, hot spots illustrate multiple success models that can serve as benchmarks for replication. Third, this study demonstrates the value of combining dynamic high-resolution sampling with spatial statistics to guide more effective and place-specific improvements in public transport.

Phase transitions during inflation can generate a stochastic gravitational-wave background that probes primordial physics. We study the detectability and parameter reconstruction of such a signal with a space-based gravitational-wave detector. Using a Taiji-like mission as a benchmark, we construct a realistic data-analysis framework that includes instrumental noise, astrophysical foregrounds and backgrounds, and the $A$, $E$, and $T$ time-delay interferometry channels. The target signal is described in a minimal, model-independent form and analyzed using both Fisher-matrix forecasts and Bayesian inference with nested sampling. We quantify detection significance and parameter-recovery thresholds, showing that while detection is achievable at moderate signal-to-noise ratios, stronger signals provide more reliable parameter reconstruction. These results offer a realistic assessment of the capability of future space-based missions to probe phase transitions during inflation through stochastic gravitational radiation.

We present a new way to construct analytical solutions for flow in complex microfluidic channel networks, as well as planar disordered media. Using a combination of Schwarz-Christoffel maps and segmentation techniques inspired by integrated circuit analysis, we build a library of base building blocks which can be reassembled to model complex geometries, in the style of ``Lego Blocks''. Our approach requires minimal numerical computation, and can then generate analytical solutions for any combination of inlet and outlet flow rates. Moreover, our method can tackle multiply connected domains which are usually difficult to model using typical conformal transform approaches. The solutions are developed for microfluidic Hele-Shaw cell devices, but also apply to ideal flow and Darcy flow in complex geometries, or any other flow problem adequately modeled by Laplace's equation. We end by showing how the procedure can be used to model complex disordered media, fractal-like flow geometries, as well as problems of steady advection-diffusion in microfluidic mixers.

Progress on the conjecture of Banica and Bichon that the classical permutation group is a maximal quantum subgroup of the quantum permutation group remains limited to a handful of small-parameter results. By Tannaka--Krein duality, any counterexample to this Maximality Conjecture must arise from a category strictly intermediate between the category $\mathcal{NC}$ of non-crossing partitions and the category $\mathcal{P}$ of all partitions. Any such exotic category must therefore contain a linear combination of crossing-partition vectors. The categories generated by $\mathcal{NC}$ together with some such vectors are studied, with a number of generation results. It is shown that no exotic category can contain a linear combination of three crossing-partition vectors, and, at $N=6$, there is no exotic category containing a linear combination of 31 crossing-partition vectors that is distinguished from $\mathcal{NC}$ or $\mathcal{P}$ at moments of order six.

This paper presents numerical values for auxiliary integrals and coefficients of the beta function in the three-loop approximation for a four-dimensional model with a quartic interaction, using a special type of regularization function. The values are compared to previously obtained results.

In the absence of error correction, noisy intermediate-scale quantum devices are operated by training parametrized quantum circuits (PQCs) so as to minimize a suitable loss function. Finding the optimal parameters of those circuits is a hard optimization problem, where global guarantees are known only for highly structured cases of limited practical relevance, and first-order methods can fail to find even local minima due to the presence of barren plateaus. In this work, we study the training of practical classes of PQCs, namely polynomial-depth circuits with a constant number of trainable parameters. This captures widely used PQC families, including fixed-depth QAOA, hardware-efficient ansätze, and Fixed Parameter Count QAOA. Our main technical result is a fully polynomial randomized approximation scheme (FPRAS), which, for every $ε>0$, returns an $ε$-approximate solution to the problem's global optimum with high probability, and has runtime and query complexity polynomial in $1/ε$ and the number of qubits. Unlike the standard hybrid quantum-classical training loop in variational algorithms, where the quantum device is queried repeatedly throughout the training, our approach separates the computation into two distinct stages: (1) an initial quantum data-acquisition phase, followed by (2) a classical global-optimization phase based on the trigonometric moment/sum-of-squares hierarchies. Under a standard flat-extension condition, which can be checked numerically, the method also supports the extraction of optimal circuit parameters. The existence of an FPRAS implies that the promise problem associated with the optimization of poly-depth constant-parameter PQC is in BQP. This imposes a limitation on the expressive power of the class, namely, it cannot encode combinatorial optimization problems whose objective values are separated by an inverse-polynomial gap.

We report Raman optical activity in pyrite FeS$_2$, which hosts an electric toroidal octupolar symmetry. A clear and reproducible sign reversal of the circular intensity difference is observed between neighboring $\{111\}$ faces under cross-circular polarization. The signal appears only for the doubly degenerate $E_g$ phonon mode and is absent for other modes, consistent with symmetry analysis. First-principles calculations reproduce these features, establishing Raman optical activity as a probe of higher-rank axial multipolar symmetry.

Spin currents can be generated through various mechanisms, including the piezospintronic effect, which arises when strain or lattice distortions induce a change in the dipolar spin moment, causing a pure spin current without necessarily being accompanied by net charge transport. This opens new possibilities for low-power information processing and novel device architectures. In this work, we propose a novel effect, the spintronic-magneto-impedictive effect, as the theoretical basis for a pure spin-current memory-like device based on antiferromagnetic components. We focus on materials that can be modeled by the so-called spin-Rice-Mele Hamiltonian, incorporating a magnetic field gradient that explicitly breaks inversion symmetry. Our results shed light on how spin currents are generated and controlled, providing new insights into the potential of these materials for next-generation spintronic technologies.

The fields of HCI and Participatory design have been turning to care ethics as a suitable ethos to approach current polycrisis with. Similar calls for relationality can be witnessed in public administration research and practice, albeit its current logic being built on privatisation and marketisation of services, managerialism and customer-focus; all of which are challenging to combine with care ethics. In this paper I use collaging technique to visually reflect on new ways for public services to adopt and (care-fully) scale participatory design approaches, and how do feminist care ethics fit in the design of public services, where there is a strong presence of neoliberalism.

Integrated visible photonic engines for solid-state quantum defects provide a foundation for scalable quantum networks. While miniaturization is advancing, active manipulation remains limited by the difficulty of achieving simultaneous milliwatt-scale visible light generation and high-contrast modulation. Despite extensive efforts, the concurrent chip-scale realization of nonlinear frequency conversion and fast temporal gating for high-fidelity quantum control has remained elusive. Here, we demonstrate a monolithic thin-film lithium niobate (TFLN) platform integrating periodically poled frequency conversion with GHz-bandwidth electro-optic (EO) switching. The device delivers off-chip green-light power exceeding 1 mW with an extinction ratio (ER) of 42.2 dB, enabling coherent spin control and time-resolved lifetime measurements of individual nitrogen-vacancy (NV) centers in diamond through nanosecond gating. System performance is validated through pulsed optically detected magnetic resonance (ODMR), Rabi oscillations, and Ramsey interference, supported by time-tagged photon counting with nanosecond resolution. By unifying sufficient nonlinear light generation with high-speed active manipulation, this platform establishes a scalable framework for the realization of high-rate quantum communication nodes.

It is thought that, sometime after their formation, the solar system's giant planets experienced a dynamical instability that caused their orbits to excite, diverge, and ejected one or more objects with masses comparable to the ice giants. A key feature of this model is that the planets experience encounters with other planetary bodies, and these encounters facilitate the capture of nearby small bodies as irregular satellites. Instability simulations indicate that planet-planet encounter distances can typically fall below 0.1 au, which is only roughly an order of magnitude larger than the radial extent of the modern planets' regular satellite systems. In this paper we model the effects of these encounters on the dynamical stability of the regular moons of Jupiter and Uranus. We tested encounter histories from 122 plausible outer solar system dynamical histories. We find that the survival probability for the Jovian and Uranian moon systems are both less than 15%. Moreover, we only identify one case where both Uranus and Jupiter's large satellites consistently survive the same instability. Interestingly, Jupiter's moons are most likely to survive in instabilities initialized with two smaller extra ice giants, and cases with one larger additional planet provide more favorable conditions for Uranian system survival. In either case, if Uranus encounters another ice giant at D<0.02 au, or one of the gas giants at D<0.1 au, satellite system destruction is effectively guaranteed. Wider encounters can also affect the system, particularly when they occur successively. Since the Laplace resonance likely would not be in place today if Jupiter's moons experienced an instability that led to collisions, our results indicate that Uranus' moons were likely perturbed to the point of collisions at least twice: as a result of both the impact that tilted the planet and the giant planet instability.

This work develops a multivariate extension of the Fixed Rank Kriging (FRK) framework for spatial prediction in settings where multiple spatial processes may provide complementary information. The goal is to preserve the computational efficiency, the ability to operate without assuming stationarity over the domain, and the spatial support flexibility of FRK, while incorporating cross-process dependence. To this end, we employ a multiresolution coregionalization structure for the latent spatial effects, in which spatial basis functions are combined with Gaussian Markov Random Field coefficients. An estimation procedure based on the expectation-maximization algorithm is developed, designed to exploit the multiresolution latent structure. Through simulation studies, we examine when the proposed joint modeling is beneficial. We consider cases in which one process is observed more sparsely or is entirely unobserved in a subregion and find that the multivariate formulation is able to borrow information from the more densely observed process, producing coherent and accurate predictions even where direct observations are limited or absent. Finally, the model is applied to the analysis of PM10 concentrations in Northern Italy, illustrating its applicability in a real environmental context.

"Synchronization of two dynamical systems" is the term used to describe the phenomenon when two or more systems gradually change their states or behaviors to become similar or identical. This can happen in a lot of fields, such as physics, engineering, biology, and economics. Synchronization finds applications in neurology and communication systems. It is present in both man-made and organic systems. The nonlinear control synchronization technique for fractional-order time derivative systems is described in this article, where the Adams Basford Moulton method is used for solving the fractional-order system. The reliability and ease of applicability for two chaotic systems are demonstrated by the numerical simulation. Furthermore, in this article, both systems were kept in a chaotic condition while being synchronized with each other. The effects of synchronizing time and rearranging the derivatives are the most significant sections of this article.

Band structure engineering in surface acoustic wave (SAW) metamaterials could advance both classical telecommunications and quantum information processing. However, no imaging technique has demonstrated the necessary capability to resolve sub-$μ$m traveling SAWs across wide GHz bandwidths. Existing methods capture only fragments of the dispersion at discrete frequencies, preventing systematic characterization and control of SAW-based metamaterials. Here, we develop electrostatic force microscopy (EFM) to enable real-space imaging of traveling SAWs in honeycomb metamaterials on LiNbO$_3$. Our application leverages sub-200 nm spatial resolution, broad GHz bandwidth, and non-contact imaging to map complex band structures with continuous frequency resolution and expanded frequency range, while preserving sub-lattice detail. Using EFM, we map the full relevant frequency range around the Dirac point of a SAW graphene analog, including the acoustic Dirac cones, and the transition from ballistic to diffusive SAW transport regime. Furthermore, by breaking sublattice symmetry, we tune the opening of a band gap at the Dirac point, and image frequency-dependent wave localization on sublattice sites. Our EFM technique closes the loop between design and real-space validation, streamlining the engineering of arbitrary SAW landscapes for next-generation applications spanning telecommunications, microfluidics, and quantum acoustics.

In recent years the theory of Higher Dimensional Automata (HDA) has seen significant advances from a theoretical point of view, reflecting standard automata theory. There have also been first attempts to use the mathematical framework provided by HDAs to known problems, in particular Petri Net analysis. However real-world applications are still lacking and issues from real-world system, as concurrency, is still opened in the context of controller generation. In this work we show how the framework of HDAs can be adapted to help transforming controllers given as interpreted Petri nets (IPN) into an actual closed loop controller and how the HDA helps in identifying ambiguous or even contradictory specifications that remain hidden in the IPN. We demonstrate the feasibility by connecting the obtain controller to a virtual environment for closed loop control, exemplified by an industrial example.

Recent observational efforts using imaging atmospheric Cherenkov telescopes (IACTs) have led to firm detections of very-high-energy (VHE) signals from bright gamma-ray bursts (GRBs), often at moderate redshifts. This work presents 15 years of H.E.S.S. GRB observations and examines their implications through population comparisons and selected modelling cases. GRBs are a key science target of the High Energy Stereoscopic System (H.E.S.S.). With a low-energy threshold ($\lesssim$100 GeV) and rapid repointing capabilities, H.E.S.S. can begin follow-up observations within tens of seconds after a GRB trigger, covering the late prompt or early afterglow phases. We report GRB follow-up observations with H.E.S.S. from 2004 to 2019, which resulted in no significant VHE signals (aside from the detections of GRB~180720B and GRB~190829A). The resulting upper limits comprise the largest set available for GRBs at VHE. A subset of bursts with favourable conditions were selected for X-ray analysis and emission modelling. Population studies were performed to compare detected and non-detected GRBs. The results indicate that VHE-detected GRBs are not a distinct population, but tend to feature luminous X-ray emission and favourable redshift and observing conditions. This highlights the potential of next-generation IACTs such as the Cherenkov Telescope Array Observatory (CTAO), whose lower energy threshold will enhance the detection of fainter and more distant GRBs.

We prove an asymptotic formula for the second moment of the first derivative of quadratic twists of modular $L$-functions with three leading order main terms. It improves the previous result of Kumar et al. with the first main term. The proof is based on the large sieve type inequality established by Li, with a key input that we convert the problem into computing an asymptotic formula for the completed twisted modular $L$-functions with large shifts.